Abstract
We investigate the future evolution of the Nariai black hole, which is the extremal limit of the Schwarzschild--de Sitter one in modified gravity. The perturbation equations around the Nariai black hole are derived in static and cosmological patches for general $F(R)$ gravity. The analytical and numerical study of several realistic $F(R)$ models shows the occurrence of a rich variety of scenarios: instabilities, celebrated Hawking evaporation, and anti-evaporation of black holes. The realization of a specific scenario depends on the model under consideration. It is remarkable that the presence of such primordial black holes in the current Universe may indicate modified gravity, which supports the anti-evaporation as the preferable model. As a generalization, we extend the study of the Nariai black hole evolution to modified Gauss-Bonnet gravity. The corresponding perturbation equations turn out to be much more complicated than in the case of $F(R)$ gravity. For a specific example of modified Gauss-Bonnet gravity, we demonstrate that the Nariai solution may be stable.